Let $k$ be an algebraically closed field of characteristic zero. Let $X$ be an affine, simplicial toric variety over $k$. If $X$ has dimension one, then it is the affine line over the field $k$, so its divisor class group is zero. Since its Chow group of 0-dimensional cycles is isomorphic to its divisor class group, we have $A_0(X)=0$. Is it true that for any $X$, we always have $A_0(X)=0$?
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$\begingroup$ If $X$ is a projective toric variety then $A_0(X) = \mathbb{Z}$ (this is true because it's resolution is rationally connected), and then the localization exact sequence implies that for any nontrivial open subset $A_0(U) = 0$. $\endgroup$– Evgeny ShinderCommented Apr 21 at 20:40
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$\begingroup$ @Evgeny Shinder Thank you very much for your kind guidance and help. $\endgroup$– BorisCommented Apr 22 at 21:08
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